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Chinese Physics B

Quantum logic gates with two-level trapped ions beyond Lamb–Dicke limit

Zheng Xiao-Juan, Luo Yi-Min, Cai Jian-Wu

Published Mar 17, 2009• 3 citations
Publisher

Abstract

In the system with two two-level ions confined in a linear trap, this paper presents a simple scheme to realize the quantum phase gate (QPG) and the swap gate beyond the Lamb–Dicke (LD) limit. These two-qubit quantum logic gates only involve the internal states of two trapped ions. The scheme does not use the vibrational mode as the data bus and only requires a single resonant interaction of the ions with the lasers. Neither the LD approximation nor the auxiliary atomic level is needed in the proposed scheme. Thus the scheme is simple and the interaction time is very short, which is important in view of decoherence. The experimental feasibility for achieving this scheme is also discussed.

DOI
10.1088/1674-1056/18/4/011

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In the system with two two-level ions confined in a linear trap, this paper presents a simple scheme to realize the quantum phase gate (QPG) and the swap gate beyond the Lamb–Dicke (LD) limit. These two-qubit quantum logic gates only involve the internal states of two trapped ions. The scheme does not use the vibrational mode as the data bus and only requires a single resonant interaction of the ions with the lasers. Neither the LD approximation nor the auxiliary atomic level is needed in the proposed scheme. Thus the scheme is simple and the interaction time is very short, which is important in view of decoherence. The experimental feasibility for achieving this scheme is also discussed.
In the system with a two-level ion confined both in a linear trap and in a high-Q single-mode cavity, we present a simple scheme to realize the basic two-qubit logic gates such as the quantum phase gate (QPG), the SWAP gate and the controlled-NOT (CNOT) gate beyond the Lamb–Dicke (LD) limit. We realize the three kinds of two-qubit quantum phase gates, i.e. QPG operation involving the cavity mode as well as the vibrational mode of the trapped ion, QPG operation involving the internal states as well as the vibrational mode of the trapped ion and QPG operation involving the internal states of the trapped ion as well as the cavity mode. The controlled-NOT gate can be implemented from a QPG operation through a rotation of the second qubit before and after the QPG operation. We can also perform the SWAP gate operation involving the ionic internal states of the trapped ion and the two-mode bosonic basis. The logic gates involving the cavity mode as well as the vibrational mode of the trapped ion are insensitive to spontaneous emission, and the logic gates involving the internal states as well as the vibrational mode of the trapped ion are insensitive to the decay of the cavity, which is an important feature for the practical implementation of quantum computing. Neither the LD approximation nor the auxiliary atomic level is needed in our scheme. Experimental feasibility for achieving our scheme is also discussed.
We propose a simple scheme for implementing quantum logic gates with a string of two-level trapped cold ions outside the Lamb-Dicke limit. Two internal states of each ion are used as one computational qubit (CQ) and the collective vibration of ions acts as the information bus, i.e., the bus qubit (BQ). Using the quantum dynamics for the laser-ion interaction as described by a generalized Jaynes-Cummings model, we show that quantum entanglement between any one CQ and the BQ can be coherently manipulated by applying classical laser beams. As a result, universal quantum gates, i.e., the one-qubit rotation and two-qubit controlled gates, can be implemented exactly. The required experimental parameters for the implementation, including the Lamb-Dicke (LD) parameter and the durations of the applied laser pulses, are derived. Neither the LD approximation for the laser-ion interaction nor the auxiliary atomic level is needed in the present scheme.
We propose a scheme to implement two-qubit controlled quantum phase gate(CQPG) via a single trapped two-level ion located in the standing wave field of a quantum cavity, in which the trap works beyond the Lamb–Dicke limit. When the light field is resonant with the atomic transition |g > ↔ |e> of the ion located at the antinode of the standing wave, we can perform CQPG between the internal and external states of the trapped ion; while the frequency of the light field is chosen to be resonant with the first red sideband of the collective vibrational mode of the ion located at the node of the standing wave, we can perform CQPG between the cavity mode and the collective vibrational mode of the trapped ion. Neither the Lamb–Dicke approximation nor the assistant classical laser is needed. Also we can generate a GHZ state if assisted with a classical laser.
We present an alternative scheme to exactly implement one-qubit and two-qubit quantum gates with a single trapped cold ion driven by a travelling laser field. The internal degree of freedom of the ion acts as the target qubit and the control qubit is encoded by two Fock states of the external vibration of the ion. The conditions to realize these operations, including the duration of each applied laser pulse and Lamb-Dicke parameter, are derived. In our scheme neither the auxiliary atomic level nor the Lamb-Dicke approximation is required. The multiquantum transition between the internal and external degrees of freedom of the ion is considered.
Quantum controlled-NOT (CNOT) gates with single trapped cold ions have been experimentally demonstrated [C. Monroe et al., Phys. Rev. Lett. 75, 4714 (1995)] by using auxiliary atomic levels (besides the qubit ones) and under the usual Lamb-Dicke (LD) limit (wherein the so-called LD parameter, which represents the interaction strength between the internal and external degrees of freedom of the driven ion, is required to be sufficiently small). It has been shown [C. Monroe et al., Phys. Rev. A 55, R2489 (1997); L. F. Wei et al., 65, 062316 (2002)] that, beyond the LD limit, these gates could also be implemented for certain specific LD parameters. Here, we further show that, without any auxiliary atomic level, the desirable quantum logic gates could still be realized with sufficiently high fidelities for arbitrary LD parameters, as long as the durations of the applied laser pulses are accurately set within the decoherence times of the ions.
We study the realization of quantum logic gates in the interacting system of a two-dimensional trapped ion and two laser beams. We simplify the Hamiltonian of the system by Choosing the frequency detuning of laser fields and the transition frequency between the internal states of the ion, and furthermore，we describe the realization of a C-NOT gate, a swap gate and a phase gate. During these processes, the system needs to meet the Lamb-Dicke limit, and the Rabi oscillation frequency should be far smaller than the phonon frequency.